A sum operator with applications to self-improving properties of Poincaré inequalities in metric spaces

Journal of fourier analysis and applications (5) 9(2003), 511–540. A SUM OPERATOR WITH APPLICATIONS TO SELF–IMPROVING PROPERTIES OF POINCAR´ E INEQUALITIES IN METRIC SPACES BRUNO FRANCHI, CARLOS P´ EREZ, AND RICHARD L. WHEEDEN Abstract. We define a class of summation operators with applications to the self-improving nature of Poincar´e–Sobolev estimates, in fairly general quasimetric spaces of homogeneous type. We show that these sum operators play the familiar role of integral operators of potential type (e.g., Riesz fractional integrals) in deriving Poincar´e–Sobolev estimates in cases when representations of functions by such integral operators are not readily available. In particular, we derive norm estimates for sum operators and use these estimates to obtain improved Poincar´e– Sobolev results. 1. Introduction. It is well-known that Poincar´e–Sobolev estimates in Euclidean space can be derived as corollaries of norm inequalities for Riesz fractional integral operators. For example, the classical estimate (1) ZB |f(x)−fB|qdx1/q ≤cZB |∇f(x)|pdx1/p ,1 q=1 p−1 n,1< p < n, where Bis a Euclidean ball in Rnand fB=1 |B|RBf(x)dx, can be derived from the norm inequality (2) ZRn |I1f(x)|qdx1/q ≤cZRn |f(x)|pdx1/p for the same values of pand q, where cis independent of fand I1f(x) = ZRn f(y) |x−y|n−1dy is the Riesz transform of fof order 1. Similarly, although (2) is false in case p= 1 and q=n/(n−1), the case p= 1 of (1) can be derived from the following weak-type analogue of (2): {x∈Rn:|I1f(x)|> λ} (n−1)/n ≤c λ||f||L1(Rn), λ > 0, with cindependent of λand f, where |E|denotes the Lebesgue measure of a set E. The well-known pointwise representation inequality |f(x)−fB| ≤ c I1(|∇f|χB)(x), x ∈B, with cindependent of x, B and f, makes it clear how (1) follows from (2) in case p > 1, and a far less obvious argument based on truncation can be used when p= 1 (see [23], [18], [28]). In fact, norm estimates for more general integral transforms have been used recently to derive Poincar´e–Sobolev estimates for vector fields in fairly general settings, such as on manifolds and groups, and even on abstract metric spaces in the sense of [6]. For example, let ρ(x, y) be a metric on Rnthat is induced by a collection Xof Carnot–Carath´eodory vector fields, and suppose that Lebesgue measure is a doubling measure for ρ-balls, i.e., that |B(x, 2r)| ≤ C|B(x, r)|with C 1991 Mathematics Subject Classification. 46E35. Key words and phrases. Poincar´e–Sobolev estimates. B.F. is supported by University of Bologna funds for selected research topics, and by GNAMPA of INdAM, Italy. 1 2 BRUNO FRANCHI, CARLOS P´ EREZ, AND RICHARD L. WHEEDEN independent of xand r, where B(x, r) denotes the ρ-ball with center xand radius r. Then the operator I(f)(x) = ZRn f(y)ρ(x, y) |B(x, ρ(x, y))|dy has known mapping properties from Lpto Lqwith p, q related naturally in terms of the doubling property, and these mapping properties lead to Poincar´e–Sobolev estimates of the form 1 |B|ZB |f(x)−fB|qdx1/q ≤c r(B)1 |B|ZB |Xf(x)|pdx1/p , where r(B) is the radius of the ρ-ball B. The reason why Poincar´e–Sobolev estimates follow is that there is a representation inequality of the form (3) |f(x)−fB| ≤ c I(|Xf|χB)(x), x ∈B, with cindependent of x, B and f; see e.g. [11], [15], [20] for precise statements of this representation, and see e.g. [28] for the mapping properties of the operator I. On the other hand, starting with work of Saloff-Coste [27], it is known that Poincar´e–Sobolev estimates have a self-improving nature, in the sense that it is possible to derive estimates for general p, q from particular special cases such as (4) 1 |B|ZB |f(x)−fB|dx ≤c r(B)1 |B|ZB |Xf|p0dx1/p0 for some p0, without explicit mention of any integral operator at all. A partial explanation for the apparent mystery about the role of integral operators in the self-improving technique was given in [11] and with sharp constants in [21] (see also [15], [20], [22], [25]). It was shown there that in case p0= 1, (4) is in fact equivalent to (3). In particular, by assuming (4) with p0= 1, we also have (3), and the more general Poincar´e–Sobolev estimates then follow from the corresponding norm estimates for the integral operator I. However, in case p0>1, no sharp representation analogous to (3) is known to follow from (4). When p0>1, the difficulty that one encounters in trying to adapt the arguments which lead from (4) to (3) in case p0= 1 is related to the presence of the exponent 1/p0: the functional a(B) defined by a(B) = r(B)1 |B|ZB |g|p0dx1/p0 (gand p0fixed) is not easy to add over a class of non-overlapping (or even disjoint) balls Bif p0>1. Thus, starting from an estimate of the type 1 |B|ZB |f−fB|dx ≤c a(B) for all balls Bwith a(B) as above and p0>1, or with an even more general functional a(B), it is not clear how to build an integral operator whose norm estimates imply improved Poincar´e– Sobolev estimates like 1 |B|ZB |f−fB|qdx1/q ≤C a(B) for some q > 1. The main purpose of this paper is to show that the familiar role of integral operators is instead played by a sum operator T(x) which is formed by adding a(B) over an appropriate chain of balls associated with a point x: T(x) = X Bin a chain for x a(B). A SUM OPERATOR WITH APPLICATIONS TO SELF–IMPROVING PROPERTIES 3 In case p0= 1, the sum operator becomes an integral operator, but in any case, the Lpto Lq mapping properties of the sum operator can be derived in much the same ways as those for integral transforms of potential type, and these norm estimates for Tlead to correspondingly more general Poincar´e estimates. We will be able to obtain such results for a fairly general class of functionals a(B) which includes the special choice a(B) = r(B)1 |B|ZB |Xf|p0dx1/p0 . See also [29], [17] and [8] for other types of operators which involve adding integral averages; the sums in [17] involve integral averages over annuli, while those in [29] involve averages over portions of dyadic “cubes”. On the other hand, the sums in [8] involve concentric balls centered at xthat hence have countable overlapping. In particular we improve some results obtained in [17]. We will use the sum operator to strengthen several of the self-improving results obtained in [16], and also to derive results for the weighted BV spaces defined in [2]. For example, we will prove the following result, in which we use the notation |B|ω=RBdω for the ω-measure of B. Let p0>0 and Xbe a differential operator on Rnfor which 1 |B|ZB |f−fB|dx ≤c r(B)1 |B|ZB |Xf|p0dx1/p0 for all ρ-balls Band all Lipschitz functions f. If ωis a measure which satisfies the doubling condition |B|ω≤cr(B) r(˜ B)N |B|ω,˜ B⊂B, for all ρ-balls ˜ B, B, then we have (5) 1 |B|ωZB |f−fB|qdω1/q ≤C r(B)1 |B|ZB |Xf|pdx1/p with p, q related by 1 q=1 p−1 N, p0≤p < q < ∞, and with Cindependent of fand B. It was proved in [16] that such a result holds under the stronger assumption that ω∈A∞(dx) (the definition of A∞(dx) is given after Corollary 2.15), but we will be able to deduce it by assuming only the doubling condition. In fact, a more general result is proved in Corollary 2.16 below, replacing (5) by (6) 1 |B|ωZB |f−fB|qdω≤C r(B)1 |B|vdx ZB |Xf|pv dx1/p . In this version, Lebesgue measure dx on the right side of the conclusion is replaced by a more general measure vdx, provided that p0≤p < q < ∞, that we replace our assumption about the doubling condition of order Nby the balance condition (7) r(˜ B) r(B) |˜ B|ω |B|ω!1/q ≤C |˜ B|vdx |B|vdx !1/p ,˜ B⊂B, and provided v∈Ap/p0(dx) (again, the definition of Ap(dx) is given after Corollary 2.15). Note that the possibility of choosing q=pis not addressed in the result just mentioned. However, in §3, we will show that if ωis absolutely continuous with respect to Lebesgue measure, it is possible to treat the case q=p>p0≥1 by assuming a stronger version of the balance condition, a version which we shall refer to as a Fefferman–Phong type strengthening of the 4 BRUNO FRANCHI, CARLOS P´ EREZ, AND RICHARD L. WHEEDEN condition; if dω =w dx, this strengthening involves replacing |˜ B|ωin the numerator on the left side of (7) by the larger quantity Ar(ω, ˜ B) = Z˜ B wrdx1/r |˜ B|1/r0 for some r > 1, 1/r + 1/r0= 1. See §3 for the exact statements. We shall refer to inequalities like (6) as two-measure (or two-weight) inequalities. To illustrate the general interest of two-weight inequalities in applications, consider the paper [5], where the authors prove a Harnack inequality for anisotropic degenerate/singular elliptic equations of the form div (A(x)Du) = 0 in an open set Ω, when λ(x)|ξ|2≤ hA(x)ξ, ξi ≤ Λ(x)|ξ|2 for ξ∈Rnand a.e. x∈Ω. There, a two-weight inequality for the pair of measures Λ dx and λ dx is a key tool used in the proof, and it is obtained directly from a balance condition akin to (7). Clearly, enlarging the class of weights for which (6) holds would yield Harnack inequalities for more general classes of pde’s. Following the spirit of [5], the same condition is used in [14] to prove a compensated compactness theorem and then a homogenization result for nonlinear degenerate elliptic pde’s with oscillating coefficients. Analogously, compact imbedding of weighted Sobolev and BV spaces can be deduced from two-weight Sobolev-Poincar´e inequalities (see, e.g., [13]). But two-weight inequalities also arise when dealing with isotropic equations of the form div (w(x)Du) = 0 in case wdoes not belong to the class A2(the situation for w∈A2is well-understood due to [9]), but when nevertheless ωcan be estimated from below and from above by weights satisying a two-weight Sobolev-Poincar´e inequality. An assumption of this type is much weaker than the A2-condition which requires more delicate control of the weight on every ball. Finally, we note that our motivation for deriving results in rather general quasimetric spaces is that the theory then works in important non-Euclidean settings like Carnot–Carath´eodory metric spaces associated with subelliptic differential operators, graphs and fractal sets (see e.g. [17] for references). 2. Main results and proofs. Throughout the paper, we shall consider a fixed quasimetric space (S, ρ) endowed with a doubling Borel measure µthat makes (S, ρ, µ) a quasimetric space of homogeneous type in the sense that the following properties hold: (i) ρ(x, y)≥0 for all x, y ∈ S, and ρ(x, y) = 0 iff x=y; (ii) ρ(x, y) = ρ(y, x) for all x, y ∈ S; (iii) ρ(x, y)≤Kρ(x, z) + ρ(z, y)for all x, y, z ∈ S. If x∈ S and r > 0, let B(x, r) denote the ρ-ball centred at xof radius r, i.e., B(x, r) = {y∈ S : ρ(x, y)< r}. If Bis a ρ-ball, we will often call Bsimply ‘a ball’, and we will denote its radius by r(B) and its µ-measure by |B|µ. Moreover, if c > 0, we shall denote by cB the ball with the same center as Band such that r(cB) = cr(B). Whenever we speak of a “measure”, we mean a nonnegative Borel measure. We always assume that the following doubling property holds for µ: (iv) There exists A > 0 such that |B(x, 2r)|µ≤A|B(x, r)|µ for all x∈ S and r. A SUM OPERATOR WITH APPLICATIONS TO SELF–IMPROVING PROPERTIES 5 Definition 2.1. We say that a locally finite Borel measure ωbelongs to the class D=D(S, ρ) if there is a constant Aω>1 so that ωsatisfies the doubling condition (8) |B(x, 2r)|ω≤Aω|B(x, r)|ω for all x∈ S and r > 0, where we denote |E|ω=REdω for any measurable set E. In case ωis absolutely continuous with respect to µ, i.e., if dω =w dµ for a nonnegative function w∈Lloc(dµ), we write |E|ω=|E|wdµ and call wa weight function. Remark 2.2.It is easy to see that (8) implies (9) |B(x, tr)|ω≤Aωtlog2Aω|B(x, r)|ω for t > 1, r > 0 and x∈ S. We shall say that ωsatisfies the doubling condition of order Nand write ω∈DN=DN(S, ρ) if |B(x, tr)|ω≤C tN|B(x, r)|ω for t > 1, r > 0 and x∈ S. If ω∈D, then by [30], p. 269, assuming as we shall that all annuli B(x, R)\B(x, r) with 0 < r < R are nonempty, ωalso satisfies a reverse doubling condition: there exist α, β > 1 depending on Aωsuch that (10) |B(x, αr)|ω≥β|B(x, r)|ω for all r > 0, and hence (11) |B(x, tr)|ω≥c t|B(x, r)|ω for all x∈ S and t > 1, where and care positive constants depending on αand β. We will usually be dealing only with the class of subballs of some fixed ball B0, and then we only need the conditions above for such balls. GEOMETRIC HYPOTHESES: Let B0be a fixed ball in (S, ρ). We suppose that for each x∈B0, there exists a chain of balls {Bj}={Bj(x)}∞ j=1 satisfying (H1) Bj⊂B0for all j≥0; (H2) r(Bj)≈2−jr(B0) for all j≥0; (H3) ρ(Bj, x)≤cr(Bj) for all j≥0, where ρ(Bj, x) denotes the distance from xto Bj, and we assume that the constants in (H2) and (H3) are independent of xand j. Note that the balls Bj(x) may or may not contain x, but the sequence {Bj(x)}depends on x. From now on, any positive constant that depends at most on K,Aand the constants in (H2) and (H3) will be called a geometric constant. It follows from (H2), (H3) and (iii) that (H4) If j < k then Bk⊂CBj, where Cis a geometric constant. Remark 2.3.We know from [20] and [15] that a chain of balls satisfying (H1)–(H3), and so also (H4), exists in metric spaces satisfying the segment (or geodesic) property, i.e., in metric spaces such that for every pair of points x, y ∈ S there is a continuous curve γ: [0, T]→ S connecting xand ysuch that ρ(γ(t), γ(s)) = |t−s|for all s, t ∈[0, T ]. In fact, we then also have the extra properties (H5) For all j≥0, Bj∩Bj+1 contains a ball Sjwith r(Sj)≈r(Bj); (H6) ρ(Bj, x)≈r(Bj) for all j≥0; (H7) {Bj}has bounded overlaps. Moreover, the constants in (H5)–(H7) are geometric constants. Typically, Carnot–Carath´eodory and Riemannian metrics satisfy the segment property (see Remark 2.6 of [16] for references). 6 BRUNO FRANCHI, CARLOS P´ EREZ, AND RICHARD L. WHEEDEN Definition 2.4. Let a:B→a(B) be a nonnegative functional defined on balls B⊂B0. If x∈B0, let (12) T(x) = ∞ X j=0 a(Bj(x)), where {Bj(x)}∞ j=1 is a sequence of balls satisfying (H1), (H2), and (H3), and B0(x) = B0for all x∈B0. We call T(x) a sum operator associated with the functional a(B). The significance of T(x) lies in the following simple pointwise representation formula. Theorem 2.5. Suppose (H1)–(H3) and (H5) hold. Let f∈L1(B0, µ)be such that for any ball B⊂B0, (13) 1 |B|µZB |f−fB|dµ ≤c a(B), where fB=1 |B|µRBf dµ. Then for µ-a.e. x∈B0, (14) |f(x)−fB0| ≤ C T(x), where Cis a geometric constant which also depends on the constant in (13). Remark 2.6.We thank Professor G. Lu for pointing out that the conclusion of Theorem 2.5 holds with a weaker hypothesis. In fact, by using the methods of [19], the left-hand side of (13) can be replaced by 1 |B|µZB |f−fB|dµ1/ for any  > 0. In particular, it can also be replaced by the weak L1(B, µ) norm of f−fB, i.e., by kf−fBkL1,∞(B,µ)= sup λ>0 λ |B|µ |{x∈B:|f−fB|> λ}|µ. The fact that the weak norm can be substituted follows from Kolmogorov’s inequality: if 0 < q < r, then for nonnegative measurable functions g, (15) 1 |B|µZB g(x)qdµ1/q ≤r r−q1/q kgkLr,∞(B,µ), where the norm on the right is the weak Lr(B, µ) norm. Moreover, the role of the constants fBand fB0in (13) and (14) can instead be played by appropriate polynomials, and in this way, our main results have analogues for high order Poincar´e– Sobolev estimates. We refer to [20] for the definition and necessary properties of polynomials in quasimetric spaces. Remark 2.6 also applies here. We can now state one of our main results, a weak type estimate for the operator T. Theorem 2.7. Let 0< q < ∞and ω∈D. Suppose (H1)–(H3) hold, and that there exist positive constants θand cso that θ < 1and (16) X j {a(Qj)q|Qj|ω}θ≤c{a(B0)q|B0|ω}θ for all collections {Qj}of pairwise disjoint subballs of B0. Then (17) sup λ>0 λ|{x∈B0:T(x)> λ}|1/q ω≤C a(B0)|B0|1/q ω, where Cis a geometric constant which also depends on the constant in (16). A SUM OPERATOR WITH APPLICATIONS TO SELF–IMPROVING PROPERTIES 7 Remark 2.8.Since θ < 1, condition (16) implies that aand ωalso satisfy the condition (called Dqin [16]) X j a(Qj)q|Qj|ω≤c a(B0)q|B0|ω for any family {Qj}of pairwise disjoint subballs of B0. In fact, this condition is weaker than (16) and corresponds to the limit case θ= 1 (not allowed here). However, we stress again that the present results, unlike those in [16], do not require that ω∈A∞(µ). Remark 2.9.In Theorems 2.5 and 2.7, the chain {Bj(x)}is not required to satisfy either (H6) or (H7). In Proposition 2.13 below, we will give important examples of functionals a(B) which satisfy (16). Here we mention the simple special case when a(B) = r(B)1 |B|µZB gpdµ1/p for 1 ≤p < ∞and a fixed function g≥0 (e.g., g=|Xf|for some f, where Xis a differential operator). In fact, for this choice of a(B), we shall see that if µ∈DNthen (16) is valid for ω=µ, 1≤p < N, 1/q = 1/p −1/N and θ=p/q, i.e., X j a(Qj)p|Qj|p/q µ≤c a(B0)p|B0|p/q µ,1 q=1 p−1 N,1≤p < N for any family of pairwise disjoint subballs of B0. As a first consequence of Theorems 2.5 and 2.7, we shall derive the following weak selfimproving property of Poincar´e’s inequality in B0. Theorem 2.10. Let (H1)–(H3) and (H5) hold. Suppose also that ω∈Dand (16) holds for some θ < 1and some 1< q < ∞. If fis a real-valued Borel function on B0that satisfies (18) 1 |B|µZB |f−cB|dµ ≤c a(B) for every ball B⊂B0, where cBis a real number depending on Band f, then (19) sup λ>0 λ|{x∈B0:|f(x)−fB0|> λ}|1/q ω≤C a(B0)|B0|1/q ω, where fB0=1 |B0|µRB0f dµ and Cis a geometric constant which also depends on the constants in (16) and (18). We now prove Theorems 2.5, 2.7 and 2.10, beginning with Theorem 2.7. Throughout the proofs, we shall denote by c, C different positive constants which may change from place to place. Proof of Theorem 2.7. For Jto be chosen and x∈B0, write T(x) = ∞ X j=0 a(Bj(x)) = J X j=0 + ∞ X j=J+1 =I+II. Then I= J X j=0 a(Bj(x)) |Bj(x)|1/q ω· |Bj(x)|−1/q ω ≤c a(B0)|B0|1/q ω J X j=0 |Bj(x)|−1/q ω, 8 BRUNO FRANCHI, CARLOS P´ EREZ, AND RICHARD L. WHEEDEN by the one-term version of (16), namely a(B)|B|1/q ω≤c a(B0)|B0|1/q ω, B ⊂B0. By (H4), BJ(x)⊂CBj(x) if j≤J, and then by reverse doubling (11), (H2) and (9), |BJ(x)|ω≤cr(BJ(x)) r(Bj(x))  |CBj(x)|ωif j≤J ≤c2(j−J)|Bj(x)|ω. Thus J X j=0 |Bj(x)|−1/q ω≤c J X j=0 2(j−J)/q|BJ(x)|−1/q ω≤c|BJ(x)|−1/q ω, and so I≤c a(B0)|B0|1/q ω|BJ(x)|−1/q ω. Notice now by (H3) that there exists a geometric constant α > 1 such that x∈αBj(x) for all j≥0. Thus, we can write II = ∞ X j=J+1 a(Bj(x)) = ∞ X j=J+1 a(Bj(x)) |Bj(x)| 1 q−1 θq ω|Bj(x)| 1 θq −1 q ω ≤"sup B:B⊂B0,x∈αB a(B)|B| 1 q−1 θq ω#∞ X j=J+1 |Bj(x)| 1 θq −1 q ω. If j≥J+ 1, then Bj(x)⊂CBJ(x) by (H4), and consequently by (11), |Bj(x)|ω≤cr(Bj(x)) r(BJ(x)) |CBJ(x)|ωif j≥J+ 1. Since 1/(θq)−1/q > 0, it follows by using (H2) that ∞ X j=J+1 |Bj(x)| 1 θq −1 q ω≤c|BJ(x)| 1 θq −1 q ω. Letting S(x) be defined by (20) S(x) = sup B:B⊂B0,x∈αB a(B)|B| 1 q−1 θq ω, we obtain II ≤c S(x)|BJ(x)| 1 θq −1 q ω. Hence (21) T(x) = I+II ≤ca(B0)|B0|1/q ω|BJ(x)|−1/q ω+S(x)|BJ(x)| 1 θq −1 q ω. We claim that (22) T(x)≤c S(x)θha(B0)|B0|1/q ωi1−θ. If S(x) is infinite, (22) is obvious. If S(x) is finite, pick Jsuch that the two terms on the right side of (21) are comparable, i.e., so that |BJ(x)|1/(θq) ω≈a(B0)|B0|1/q ω S(x). A SUM OPERATOR WITH APPLICATIONS TO SELF–IMPROVING PROPERTIES 9 Indeed, to see that this choice is possible, let M=a(B0)|B0|1/q ω/S(x). Then 0< M ≤a(B0)|B0|1/q ω a(B0)|B0| 1 q−1 θq ω =|B0|1/(θq) ω, or equivalently |B0|ω≥Mθq. By reverse doubling, |Bj(x)|ω→0 as j→ ∞. Hence, there exists Jsuch that |BJ+1(x)|ω< Mθq and |BJ(x)|ω≥Mθq. Thus |BJ+1(x)|ω<|BJ(x)|ω. On the other hand, |BJ+1(x)|ω≈ |BJ(x)|ωsince by (H4), (23) BJ+1(x)⊂CBJ(x), and then we have |BJ(x)|ω≤ |CBJ(x)|ω≤r(CBJ(x)) r(BJ+1)N |BJ+1(x)|ω≤c|BJ+1(x)|ω, where the next-to-last inequality follows from (23) and doubling applied to the balls BJ+1(x) and CBJ(x), and the last inequality follows from (H2). Hence |BJ(x)|ω≈Mθq, as desired, and we then obtain (22) by direct computation. If T(x)> λ, then (22) implies that λ < c S(x)θha(B0)|B0|1/q ωi1−θ. Hence, by definition of S(x), there exists a ball Bxwith Bx⊂B0,x∈αBxand λ<c a(Bx)|Bx| 1 q−1 θq ωθha(B0)|B0|1/q ωi1−θ, so that (24) λq|Bx|ω≤c a(Bx)θq|Bx|θ ωha(B0)|B0|1/q ωiq(1−θ). Since the collection of balls C={αBx:x∈B0and T(x)> λ}covers {x∈B0:T(x)> λ}, an argument of Vitali type shows that there is a disjoint countable subfamily {αBk}∞ k=1 of C(thus (24) holds for each Bk) and a geometric constant α1>1 such that {x∈B0:T(x)> λ} ⊂ ∞ [ k=1 α1αBk. Hence |{x∈B0:T(x)> λ}|ω≤ ∞ X k=1 |α1αBk|ω ≤c ∞ X k=1 |Bk|ωby doubling ≤c λq ∞ X k=1 a(Bk)θq|Bk|θ ωha(B0)|B0|1/q ωiq(1−θ), by (24). Since the balls Bkare disjoint and lie in B0, we obtain from (16) that |{x∈B0:T(x)> λ}|ω≤c λqa(B0)θq|B0|θ ωha(B0)|B0|1/q ωiq(1−θ) =c λqa(B0)q|B0|ω. 16 BRUNO FRANCHI, CARLOS P´ EREZ, AND RICHARD L. WHEEDEN Theorem 2.22. Let (S, ρ, µ)be a metric space of homogeneous type which is also of Poincar´e type (i.e., (43) holds for all f∈Liploc(S,R)), and suppose that (H1)–(H3) and (H5) hold. Let w dµ ∈Dbe such that there exists q > 1so that r(B) r(B0)|B|ωdµ |B0|ωdµ 1/q ≤C|Q|µ |B0|µ for all balls B⊂B0. Then (44) 1 |B|ωdµ ZB |f−fB|qω dµ1/q ≤Cr(B) |B|µ kDfkS(B) for all f∈BVS(B0)and B⊂B0. Remark 2.23.If we assume in addition that (S, ρ) enjoys the segment property, then both Theorems 2.19 and 2.22 could be proved alternatively through a representation formula and an Lp, Lqcontinuity result for integral operators of potential type in spaces of homogeneous type. See [20] for the form of this representation, and see e.g. [10] for the Lp, Lqcontinuity result. Similar representation formulas were introduced earlier in [11] and [15] in case a stronger assumption is satisfied by the measures involved. In the case of Theorem 2.19 for example, the assumption requires the existence of c > 0 such that for all balls B,˜ Bwith ˜ B⊂B⊂B0, |B|wdx |˜ B|wdx ≥cr(B) r(˜ B), which fails to hold for general A∗ 1weights (think for instance of w(x) = |x|−n+for 0 <  < 1). However, by [20], this stronger condition is not required if the representation formula in [11], [15] is altered slightly by adding an innocuous constant term to the right-hand side. 3. The case p=q. In this section, we shall consider the special case when the functional a(B) is given by (45) a(B) = r(B)1 |B|µZB gp0dµ1/p0 , B ⊂B0, where p0≥1, µ∈D, and g≥0. We will not need to assume that gis a derivative, but we will assume that ωis absolutely continuous with respect to µ: dω =w dµ. Our goal is to derive an analogue of Corollary 2.16 in which qis allowed to equal pif p > p0, i.e., to prove that for appropriate wand v, the estimate 1 |B0|wdµ ZB0 |f−fB0|pw dµ1/p ≤C r(B0)1 |B0|vdµ ZB0 gpv dµ1/p with p > p0can be deduced from an initial assumption of the form 1 |B|µZB |f−fB|dµ ≤C r(B)1 |B|µZB gp0dµ1/p0 for all balls B⊂B0. The exact statement is given in Theorem 3.1 below. It will be convenient to assume as we may that supp g ⊂B0, and then to define a(B) by the same formula for all B⊂ S. Let Ar(w, B) = ZB wrdµ1/r |B|1/r0 µ, r > 1,1 r+1 r0= 1. Note that |B|wdµ ≤ Ar(w, B) for any wby H¨older’s inequality, and that if w∈A∞(dµ), then |B|wdµ ≈ Ar(w, B) uniformly in Bif ris sufficiently close to 1. In this section, in order to prove A SUM OPERATOR WITH APPLICATIONS TO SELF–IMPROVING PROPERTIES 17 a direct strong type estimate for the sum operator T(x), we will assume a different form of the balance condition. We will assume that for a given p > p0, there exists r > 1 so that for all B⊂cB0(c > 1 is an appropriate geometric constant), (46) r(B) r(B0)p0Ar(w, B) |cB0|wdµ p0 pAr(σ, B) |cB0|σdµ 1−p0 p≤C|B|µ |B0|µ , σ =v−1 (p/p0)−1. We refer to this condition as a strengthened balance condition in the Fefferman–Phong sense. In case v∈Ap/p0(dµ), it is easy to check that (46) amounts to the balance condition (47) r(B) r(B0)Ar(w, B) |cB0|wdµ 1/p ≤C|B|vdµ |B0|vdµ 1/p , B ⊂cB0. Moreover, if w∈A∞(dµ), (47) is equivalent to (36) in case p=q(and ν=µ): r(B) r(B0)|B|ωdµ |B0|ωdµ 1/p ≤c|B|vdµ |B0|vdµ 1/p , B ⊂B0. We will prove the analogue of Corollary 2.16 given in the next theorem. Theorem 3.1. Assume that (H1)–(H3) and (H5) hold for a ball B0in a space (S, ρ, µ)of homogeneous type. Let fbe a function which satisfies 1 |B|µZB |f−fB|dµ ≤C r(B)1 |B|µZB gp0dµ1/p0 , B ⊂B0, for some p0≥1and some function g≥0. If wand vare a pair of weights so that the balance condition (47) holds for some p > p0,r > 1and all B⊂cB0, and if v∈Ap/p0(dµ), then ZB0 |f−fB0|pw dµ1/p ≤C|cB0|1/p wdµ r(B0)1 |B0|vdµ ZB0 gpv dµ1/p . Note that the function gabove is not assumed to be a derivative. Note also that w dµ is not assumed to be a doubling measure; if w dµ is doubling then we may take c= 1 in the conclusion. We will use the following result about sum operators as a basis for deriving Theorem 3.1. Theorem 3.2. Assume that (H1)–(H3) hold for a ball B0in a space (S, ρ, µ)of homogeneous type. Let Tbe the sum operator formed by using the functional a(B)in (45) for some p0≥1. Let wand vbe weights which satisfy (46) for some p > p0and all B⊂cB0. Then (48) ZB0 Tpw dµ1/p ≤CB0ZB0 gpv dµ1/p with CB0=Cr(B0)|cB0| 1 p wdµ|cB0| 1 p0−1 p σdµ |B0| 1 p0 µ . Remark 3.3.As always, µis assumed to be a doubling measure but none of w, v or σis assumed to be a doubling weight. If v∈Ap/p0(dµ), then (48) means simply that kTkLp wdµ(B0)≤C r(B0)|cB0|1/p wdµ 1 |B0|vdµ ZB0 gpv dµ1/p , since if v∈Ap/p0(dµ) then |cB0| 1 p0−1 p σdµ |B0| 1 p0 µ ≤C |B0| 1 p vdµ . 18 BRUNO FRANCHI, CARLOS P´ EREZ, AND RICHARD L. WHEEDEN Finally, note that Theorem 3.2 is a strong type result, as opposed to our earlier weak type result about T(x). Proof of Theorem 3.2. To prove the theorem, we will use a grid of dyadic sets in Swhich are “almost balls”, as constructed in [28]. In fact, the following has been proved there: If τ= 8K5(where Kis the quasimetric constant for ρ), then for any (large negative) integer m, there are points {xk j}and a family Dm={Dk j}of sets for k=m, m + 1,· · · and j= 1,2,· · · such that •B(xk j, τk)⊂Dk j⊂B(xk j, τk+1) •For each k=m, m + 1,· · · , the family {Dk j}is pairwise disjoint in j, and S=∪jDk j. •If m≤k < l, then either Dk j∩Dl i=∅or Dk j⊂Dl i. We call the family D=∪m∈ZDma dyadic cube decomposition of Sand refer to the sets in Das dyadic cubes. A dyadic cube will usually be denoted by Q, and B(Q) will denote the containing ball described above with 1 τB(Q)⊂Q⊂B(Q); thus, if Q=Dk jthen B(Q) = B(xk j, τk+1). We set `(Q) = r(B(Q))/τ and call `(Q) the “sidelength” of Q. We note that while the cubes in each Dmhave the dyadic properties listed above, there may be no nestedness properties of the cubes in Dm1relative to the cubes in Dm2if m1, m2are different. Since supp g ⊂B0, the theorem will follow by proving (48) with integration on the right-hand side extended over S. Let x∈B0. By definition, T(x) = Xa(B), where the sum is over all balls Bin a chain for x. Define Tm(x) = X B:r(B)≥τm a(B), where the sum is only over those balls in the same chain whose radius is at least τm. Since Tm(x) increases to T(x) as m→ −∞, it is enough to prove (48) with Treplaced by Tmfor the same constant CB0(independent of m). Fix m. If Bbelongs to the chain for xand r(B)≥τm, then if r(B)≈2−nr(B0), n≥0, we can choose pairwise disjoint dyadic cubes Qn `∈ Dm,`= 1, . . . , N, of comparable size to B(i.e., with `(Qn `)≈r(B)) such that B⊂SN `=1 Qn `. In fact, Ncan be chosen to be independent of B. If Qis a dyadic cube, let a(Q) = `(Q)1 |Q|µZQ gp0dµ1/p0 . Since r(B)≈`(Qn `), it follows from doubling that |B|µ≈ |Qn `|µ. Thus, since the Qn `are disjoint in `, there is a geometric constant cdepending possibly also on Nand p0so that a(B) = r(B)1 |B|µZB gp0dµ1/p0 ≤cX ` `(Qn `) 1 |Qn `|µZQn ` gp0dµ!1/p0 =cX ` a(Qn `). Since ρ(x, B)≤cr(B) (by (H3)), then ρ(x, Qn `)≤c `(Qn `) for all `. Hence, (49) Tm(x)≤cX `, n :Qn `∈Dm ρ(x,Qn `)≤c `(Qn `) a(Qn `). A SUM OPERATOR WITH APPLICATIONS TO SELF–IMPROVING PROPERTIES 19 By duality, kTmkLp wdµ(B0)= sup h≥0, supp h⊂B0 khkLp0 dµ(B0)=1 ZTmh w 1 pdµ, 1 p+1 p0= 1. Since B⊂B0, it is easy to see that each Qn `is contained in cB0, and consequently we obtain ZTmh w1 pdµ ≤cX Q∈Dm;Q⊂cB0 a(Q)ZcB(Q) h w1 pdµ =cX Q∈Dm;Q⊂cB0 `(Q)1 |Q|µZQ gp0dµ1 p0ZcB(Q) h w1 pdµ := c S. To prove Theorem 3.2, it is enough to show that (50) S≤CB0Zgpv dµ1 pZhp0dµ1 p0 . We may assume without loss of generality that 1 |B0|µZB0 gp0dµ1 p0= 1. For γ > 1 to be chosen and k∈Z, let (51) Ck={Q∈ Dm:Q⊂cB0;γk<1 |Q|µZQ gp0dµ1 p0≤γk+1}. Then S=X Q∈Dm;Q⊂cB0 `(Q)1 |Q|µZQ gp0dµ1 p0ZcB(Q) h w1 pdµ =X kX Q∈Ck `(Q)1 |Q|µZQ gp0dµ1 p0ZcB(Q) h w1 pdµ =X k≤0 +X k≥1 := S1+S2. Let us first estimate S1. We have S1≤X k≤0X Q∈Ck `(Q)γk+1 ZcB(Q) h w1 pdµ ≤X k≤0 γk+1 X Q∈Dm:Q⊂cB0 `(Q)ZcB(Q) h w1 pdµ. We claim that if Bis any ball, then (52) X Q∈Dm;Q⊂cB `(Q)ZcB(Q) h w1 pdµ ≤c r(B)ZcB h w1 pdµ. To prove (52), note that the left-hand side of (52) is at most X `:τ`≤cr(B)X Q∈Dm, Q⊂cB `(Q)=τ` τ`ZcB(Q) h w1 pdµ 20 BRUNO FRANCHI, CARLOS P´ EREZ, AND RICHARD L. WHEEDEN ≤X `:τ`≤cr(B) τ`ZcB  X Q∈Dm:`(Q)=τ` χcB(Q) h w1 pdµ := I. But X Q∈Dm:`(Q)=τ` χcB(Q)(y)≤C uniformly with respect to `(cf. (59) of [26]). Thus since X `:τ`≤cr(B) τ`≤c r(B), we obtain I≤Cr(B)ZcB h w1 pdµ. which proves (52). Going back to S1, we obtain from (52) that S1≤c r(B0)ZcB0 h w1 pdµ. Thus, since 1 |B0|µRB0gp0dµ = 1, we may write S1≤c r(B0)1 |B0|µZB0 gp0dµ1 p0ZcB0 h w1 pdµ =c r(B0)1 |B0|µZB0 gp0v1 sv−1 sdµ1 p0ZcB0 h w1 pdµ with s=p/p0. By H¨older’s inequality, S1≤c r(B0)1 |B0|p0 µZB0 gpv dµ1 pZB0 v−s0 sdµ1 p0s0ZcB0 hp0dµ1 p0 |cB0| 1 p wdµ ≤CB0ZB0 gpv dµ1 pZcB0 hp0dµ1 p0 . This completes our estimation of S1. To estimate S2, let {Qk j}jbe the maximal dyadic cubes in Dmwith 1 |Q|µZQ gp0dµ1 p0> γk. The Qk jare disjoint in jby maximality. We do not assume Qk j⊂cB0, but if k≥1, this must be so for a suitably large geometric constant cprovided γis large, as we now show. In fact, if Qk jis not contained in cB0and cis sufficiently large depending on the quasimetric constant K, then `(Qk j) is at least comparable to r(B0) since Qk jmust intersect B0(due to the support of g). Consequently, we must have |B0|µ≤c1|Qk j|µby doubling, with c1depending on c, and then 1 = 1 |B0|µZB0 gp0dµ1 p0=1 |B0|µZB0 gp0dµ1 p0 ≥c−1 p0 1 1 |Qk j|µZQk j gp0dµ!1 p0 ≥c−1 p0 1γk, which is impossible for k≥1 if γis sufficiently large. A SUM OPERATOR WITH APPLICATIONS TO SELF–IMPROVING PROPERTIES 21 Thus Qk j⊂cB0if k≥1. By maximality and since µis doubling we have (53) γk< 1 |Qk j|µZQk j gp0dµ!1 p0 < cγk≤γk+1, if γis large, so that Qk j∈ Ckwhen k≥1. On the other hand, again by maximality, any cube Q∈ Ckis contained in a cube Qk jfor some j. Then S2≤cX k γk+1 X jX Q∈Dm:Q⊂Qk j `(Q)ZcB(Q) h w1 pdµ. If we write Bk j=B(Qk j) and apply (52), we obtain that S2is bounded by cX k γk+1 X j `(Qk j)ZcBk j h w1 pdµ ≤cγ X k,j `(Qk j) 1 |Qk j|µZQk j gp0dµ!1 p0ZcBk j h w1 pdµ =cγ X k,j a(Qk j)ZcBk j h w1 pdµ.(54) By H¨older inequality with exponents (pr)0,pr, ZcBk j h w1 pdµ ≤ ZcBk j h(pr)0 dµ!1 (pr)0 ZcBk j wrdµ!1 pr ≤ ZcBk j h(pr)0 dµ!1 (pr)0 Ar(w, cBk j)1 p|cBk j|−1 pr0 µ. Then, by H¨older’s inequality for p, p0, (54) and so also S2is bounded by (55) cγ  X k,j a(Qk j)pAr(w, cBk j)  1 p  X k,j ZcBk j h(pr)0 dµ!p 0 (pr)0 |Qk j|−p 0 pr0 µ   1 p0 . We stress the fact that Qk j, Bk j⊂cB0, as we proved above. Note that −p0/pr0=−p0/(pr)0+ 1, so the second factor in (55) is   X k,j 1 |Qk j|µZcBk j h(pr)0 dµ!p 0 (pr)0 |Qk j|µ   1 p0 . Let Ωk={x: sup Q∈Dm:x∈Q1 |Q|µZQ gp0dµ1 p0> γk}. Then Ωk=SjQk j. Let Ek j=Qk j\Ωk+1. Note Ek j⊂Ωk\Ωk+1, and therefore the sets {Ek j}are disjoint in both kand j. We claim that (56) |Qk j|µ≤2|Ek j|µ. 22 BRUNO FRANCHI, CARLOS P´ EREZ, AND RICHARD L. WHEEDEN If so, the second factor in (55) is bounded by   X k,j 1 |Qk j|µZcBk j h(pr)0 dµ!p0/(pr)0 2|Ek j|µ   1/p0 . On the other hand, if x∈Ek jthen x∈Qk j, so that if we denote by Mthe Hardy-Littlewood maximal function defined by M(f)(x) = sup B:x∈B 1 |B|µZB |f|dµ, we obtain from the doubling of µthat 1 |Qk j|µZcBk j h(pr)0 dµ ≤cM h(pr)0(x) if x∈Ek j. Hence, the second factor in (55) is bounded by c X j,k ZEk j Mh(pr)0p0 (pr)0dµ  1 p0 (57) ≤c"ZMh(pr)0p0 (pr)0dµ#1 p0 ≤cZhp0 dµ1 p0 since p0/(pr)0>1. To prove (56), it is enough to show that |Qk j∩Ωk+1|µ≤1 2µ(Qk j). Write |Qk j∩Ωk+1|µ=|Qk j∩[ i Qk+1 i|µ =X i |Qk j∩Qk+1 i|µ. If Qk j∩Qk+1 i6=∅, then either Qk+1 i⊂Qk jor Qk j⊂Qk+1 iand Qk j6=Qk+1 i.But the last is impossible, since by maximality of Qk jit would imply that γk≥ 1 |Qk+1 i|µZQk+1 i gp0dµ!1 p0 , A SUM OPERATOR WITH APPLICATIONS TO SELF–IMPROVING PROPERTIES 23 which is false since the right-hand side exceeds γk+1. Thus Qk+1 i⊂Qk jif the two intersect, so that |Qk j∩Ωk+1|µ=X i:Qk+1 i⊂Qk j |Qk+1 i|µ ≤X i:Qk+1 i⊂Qk j 1 γ(k+1)p0ZQk+1 i gp0dµ ≤1 γ(k+1)p0ZQk j gp0dµ since the Qk+1 iare disjoint in i ≤1 γ(k+1)p0(cγk)p0|Qk j|µby (57) =c γp0 |Qk j|µ≤1 2µ(Qk j) if γis chosen sufficiently large. Thus, our claim (56) is proved. We now want to estimate the first factor in (55). Recall that a(Q) = `(Q)1 |Q|µZQ gp0dµ1 p0 and p > p0≥1. Then, writing again Bk j=B(Qk j) and setting s=p/p0, we have X k,j a(Qk j)pAr(w, cBk j)≤cX k,j r(Bk j)p 1 |Bk j|µZBk j gp0v1 sv−1 sdµ!s Ar(w, cBk j). By H¨older’s inequality with exponents (s0r)0, s0r, the last sum is bounded by cX k,j r(Bk j)p|Bk j|−s µ ZBk j gp0(s0r)0v(s0r)0 sdµ!s (s0r)0 ZBk j v−s0r sdµ!s s0r Ar(w, cBk j).(58) Remember that by definition of Ar, ZBk j v−s0r sdµ!s s0r =Arv−s0 s, Bk js s0 |Bk j|−s s0r0 µ. In addition, since s/s0=s−1 = p/p0−1, we have by (46) that if Bis any subball of cB0, then r(B) r(B0)pAr(w, B) |cB0|wdµ   Ar(v−s0 s, B) RcB0v−s0 sdµ  s s0 ≤c|B|µ |B0|µp p0. Applying this with B=cBk j, recalling that µis doubling, and writing cin place of c2as necessary, we obtain that (58) is bounded by cr(B0)p|cB0|wdµ RcB0v−s0 sdµs s0 |B0| p p0 µ (59) ·X j,k |Bk j| p p0−s−s s0r0 µ ZBk j gp0(s0r)0v(s0r)0 sdµ!s (s0r)0 . 24 BRUNO FRANCHI, CARLOS P´ EREZ, AND RICHARD L. WHEEDEN The first factor in (59) is precisely the scaling factor Cp B0appearing in (48). To estimate the second factor in (59) (i.e., the sum), note that p p0 −s−s s0r0=−s s0r0= 1 −s (s0r)0 since s (s0r)0−s s0r0=s1 (s0r)0−1 s0r0 =s1−1 s0r−1 s0r0=s1−1 s0=s1 s= 1. Therefore, the sum in (59) equals X j,k 1 |Bk j|µZBk j gp0(s0r)0v(s0r)0 sdµ !s (s0r)0 |Bk j|µ, which as before (using |Bk j|µ≈ |Qk j|µ≤c|Ek j|µ) is bounded by cZMgp0(s0r)0v(s0r)0 ss (s0r)0 dµ. Since s (s0r)0>1, the last integral is at most cZgp0(s0r)0v(s0r)0 ss (s0r)0 dµ =cZgp0sv dµ =cZgpv dµ. Combining estimates and taking the p-th root shows that the first factor in (55) is bounded by CB0Rgpv dµ1/p. Using this together with the estimate (57) for the second factor in (55), we see that that (55), and so also S2, is bounded by CB0Zgpv dµ1/p Zhp0dµ1/p0 . We have already shown that S1has the same bound, and therefore so does S, i.e., (50) holds, and the proof is complete.  Proof of Theorem 3.1. 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P. Ziemer, Weakly Differentiable Functions, Springer, 1989. Bruno Franchi: Dipartimento di Matematica, Universit` a di Bologna, Piazza di porta San Donato, 5, 40126 Bologna, Italy. E-mail address:[email protected] Carlos P´ erez: Departmento de An´ alisis Matem´ atico, Facultad de Matem´ aticas, Universidad de Sevilla, 41080 Sevilla, Spain. E-mail address:[email protected] Richard. L. Wheeden: Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08903, USA. E-mail address:[email protected]